Problem Set 6—Partial Derivatives

Purpose

This problem set reinforces your understanding of limits of multi-variable functions, and begins to develop your ability to work with partial derivatives.

Background

We discussed limits of multi-variable functions in lectures on October 14 and October 16. We will talk about partial derivatives on October 16 and 20. Section 14.2 of our textbook covers limits, while 14.3 and 14.4 cover partial derivatives (or at least the aspects of them covered by this problem set).

Activity

Solve each of the following problems:

Problem 1

Exercise 50 in section 14.2 of our textbook (show that lim_{(x,y)→(1,-1)}( (xy+1) / (x²-y²) does not exist).

Problem 2

Exercise 6 in section 14.3 of our textbook (find the partial derivatives with respect to x and y of f(x,y) = (2x - 3y)³).

Problem 3

Exercise 42 in section 14.3 of our textbook (find all the second-order partial derivatives of f(x,y) = sin(xy)).

Problem 4

Exercise 66 in section 14.3 of our textbook (find ∂x/∂z at (1,-1,-3) given that xz + ylnx - x² + 4 = 0).

Problem 5

Exercise 2 in section 14.4 of our textbook (find dw/dt given w = x² + y², x = cost + sint, and y = cost - sint; find the derivative in 2 ways and evaluate at t = 0, as described in the book).

Problem 6

Exercise 8 in section 14.4 of our textbook (find ∂z/∂u and ∂z/∂v given z = tan^-1(x/y), x = ucosv, and y = usinv; find the derivatives in 2 ways and evaluate at (u,v) = ( 1.3, π/6 ), as described in the book).

Follow-Up

I will grade this exercise in a face-to-face meeting with you. During this meeting I will look at your solution, ask you any questions I have about it, answer questions you have, etc. Please bring a written solution to the exercise to your meeting, as that will speed the process along.

Sign up for a meeting via Google calendar. If you worked in a group on this exercise, the whole group should schedule a single meeting with me. Please make the meeting 15 minutes long, and schedule it to finish before the end of the “Grade By” date above.